Note that \(J(\hat{z}) = \frac{1}{4} {\interleave \hat{z}\interleave }^4 + J_1(\hat{z})\) , where \(J_1(\hat{z})= \sum _k a_k \hat{z}_k^2\)  with \(a_k = \frac{T\Phi _k^{\prime }(0)}{\gamma \omega ^4 k^2}\) . By Lemma REF  the sequence \((a_k)_k\)  is converging to 0 as \(|k|\rightarrow \infty \) , so in particular it is bounded. Due to () one finds that \(J\)  is well defined and continuous on \({\mathcal {D}(J)}\) , and moreover, that for \(\hat{z}\in {\mathcal {D}(J)}\)
\(J(\hat{z}) \ge \frac{1}{4}{\interleave \hat{z}\interleave }^4 -\sup _k |a_k| \sum _k\hat{z}_k^2\ge \frac{1}{4}{\interleave \hat{z}\interleave }^4 - \sup _k |a_k|{\interleave \hat{z}\interleave }^2.\) 

This implies that \(J\)  is coercive and bounded from below. The weak lower semi-continuity of \(J\)  follows from the convexity and continuity of the map \(\hat{z} \mapsto {\interleave \hat{z}\interleave }^4\)  and the weak continuity of \(J_1\) . To see the latter take an arbitrary \(\epsilon >0\) . Then there is \(k_0\in \mathbb {N}\)  such that \(|a_k|\le \epsilon \)  for \(|k|> k_0\)  and this implies the inequality
\( |J_1(\hat{z})- J_1(\hat{y})| \le \sup _k |a_k| \sum _{|k|\le k_0} |\hat{z}_k^2-\hat{y_k}^2| + \epsilon (\Vert \hat{z}\Vert _{l^2}^2 + \Vert \hat{y}\Vert _{l^2}^2) \quad \forall \,\hat{z},\hat{y}\in {\mathcal {D}(J)}.\) 

Since \(({\mathcal {D}(J)},{\interleave \cdot \interleave })\)  continuously embeds into \(l^2\)  any weakly convergent sequence in \(({\mathcal {D}(J)},{\interleave \cdot \interleave })\)  also weakly converges in \(l^2\)  and in particular pointwise. This pointwise convergence together with the boundedness of the sequence and (REF ) yields the weak continuity of \(J_1\)  and thus the weak lower semi-continuity of \(J\) . As a consequence, cf. Theorem 1.2 in [1]}, we get the existence of a minimizer.
In order to check that the minimizer is nontrivial is suffices to verify that \(J\)  attains negative values. Here we distinguish between case (i) and (ii) in the assumptions of the theorem. In case (i) when \(\gamma <0\)  we find an index \(k_0\)  such that \(\Phi _{k_0}^{\prime }(0)>0\) . In case (ii) when \(\gamma >0\)  we choose \(k_0\)  such that \(\Phi _{k_0}^{\prime }(0)<0\) . In both cases we obtain that \(\Phi _{k_0}^{\prime }(0)/\gamma <0\) . If we set \(\hat{y}(\delta _{k,k_0}-\delta _{k,-k_0})_{k\in {\mathbb {Z}_{odd}}}\)  then \(\hat{y}\)  has exactly two non-vanishing entries, namely \(+1\)  at \(k_0\)  and \(-1\)  at \(-k_0\) . Hence \(\hat{y}\in {\mathcal {D}(J)}\) . Using the property \(\Phi _{k_0}^{\prime }=\Phi _{-k_0}^{\prime }\)  we find for \(t\in \mathbb {R}\)
\(J(t \hat{y})=t^4\frac{1}{4}{\interleave \hat{y}\interleave }^4 +2t^2\frac{T\Phi ^{\prime }_{k_0}(0)}{\gamma \omega ^4k_0^2}\) 

which is negative by the choice of \(k_0\)  provided \(t>0\)  is sufficiently small. Thus, \(\inf _{{\mathcal {D}(J)}}J<0\)  and every minimizer \(\hat{\alpha }\)  is nontrivial.
Next we show for every critical point \(\hat{\alpha }\)  of \(J\)  that \(w(x,t)\sum _{k\in {\mathbb {Z}_{odd}}}\frac{\hat{\alpha }_k}{k}\Phi _k({x})e_k(t)\)  is a weak solution of (REF ). The regularity properties \(w\in {H^{1}({\mathbb {R}\times T})}\) , \(\partial _tw(0,\cdot )\in {L^{4}(T)}\)  and the exponential decay have already been shown in Lemma REF . We skip the standard proof that \(J\in {C^{1}({\mathcal {D}(J)},\mathbb {R})}\)  and that its Fréchet-derivative is given by (REF ). We will show that (REF ) holds for any \(\psi \)  as in (REF ) with even functions \(\Psi _k\in H^1(\mathbb {R})\) , \(\Psi _k=-\Psi _{-k}\)  such that \(\psi _x, \psi _t \in L^2(D)\)  and \(\psi (0,\cdot )\in L^4(T)\)  as described in Lemma REF . We begin by deriving expressions and estimates for the functionals
\(H_1(\psi ) = \int _D g(x) w_t \psi _t\,d(x,t), \quad H_2(\psi ) = \int _D w_x \psi _x \,d(x,t), \quad H_3(\psi ) = \int _{0}^{T} w_t(0,t)^3\psi _t(0,t)\,dt.\) 

In a first step we assume that the sum in (REF ) is finite in order to justify the exchange of summation and integration in the following. Then, starting with \(H_1\)  we find
\(H_1(\psi )&= -\omega ^2 \int _D g(x)\sum _{k,l} \hat{\alpha }_k\Phi _k({x}) \Psi _l(|x|) e_k(t)e_l(t) {(x,t)} \\&=-2\omega ^2 \sum _k \hat{\alpha }_k\int _0^\infty g(x)\Phi _k(x)\Psi _{-k}(x){x}\\&=2\omega ^2 \sum _k \hat{\alpha }_k\int _0^\infty g(x)\Phi _k(x)\Psi _k(x){x},\\|H_1(\psi )| &\le 2\omega ^2 \Vert g\Vert _{L^\infty (\mathbb {R})} \Bigl (\sum _k \hat{\alpha }_k^2 \Vert \Phi _k\Vert ^2_{L^2(0,\infty )}\Bigr )^\frac{1}{2} \Bigl (\sum _k\Vert \Psi _k\Vert _{L^2(0,\infty )}^2\Bigr )^\frac{1}{2}= \Vert g\Vert _{L^\infty (\mathbb {R})} \Vert w_t\Vert _{L^2(D)}\Vert \psi _t\Vert _{L^2(D)}\) 

and similarly for \(H_2\)  we find using (REF )
\(H_2(\psi ) &= \int _D \sum _{k,l} \frac{\hat{\alpha }_k}{k}\Phi _k^{\prime }({x}) \frac{1}{l}\Psi _l^{\prime }(|x|) e_k(t)e_l(t) {(x,t)} \\&= 2\sum _k \frac{\hat{\alpha }_k}{-k^2} \int _0^\infty \Phi _k^{\prime }(x)\Psi _{-k}^{\prime }(x){x}\\&= 2\sum _k \frac{\hat{\alpha }_k}{k^2} \int _0^\infty \Phi _k^{\prime }(x)\Psi _k^{\prime }(x){x} \\&= 2\omega ^2\sum _k \hat{\alpha }_k \int _0^\infty g(x) \Phi _k(x)\Psi _k(x){x} - 2\sum _k \frac{\hat{\alpha }_k}{k^2}\Phi _k^{\prime }(0)\Psi _k(0), \\|H_2(\psi )| &\le 2\Bigl (\sum _k \frac{\hat{\alpha }_k^2}{k^2}\Vert \Phi _k^{\prime }\Vert _{L^2(0,\infty )}^2 \Bigr )^\frac{1}{2} \Bigl (\sum _k \frac{1}{k^2}\Vert \Psi _k^{\prime }\Vert _{L^2(0,\infty )}^2 \Bigr )^\frac{1}{2} = \Vert w_x\Vert _{L^2(D)}\Vert \psi _x\Vert _{L^2(D)}.\) 

Moreover, considering \(H_3\)  and setting \(\hat{y}_k \Psi _k(0)\)  for \(k\in {\mathbb {Z}_{odd}}\)  one sees
\(H_3(\psi ) &= \omega ^4 \int _{0}^{T}\Bigl (\sum _k \hat{\alpha }_ke_k(t)\Bigr )^3\Bigl (\sum _l \Psi _l(0)e_l(t)\Bigr ){t} \\&= \frac{\omega ^4}{T} (\hat{\alpha }*\hat{\alpha }*\hat{\alpha }*\hat{y})_0,\\{H_3(\psi )} & \le \frac{\omega ^4}{T} {\interleave \hat{\alpha }\interleave }^3 {\interleave \hat{y}\interleave }= {w_t(0,\cdot )}_{L^{4}(T)}^3{\psi _t(0,\cdot )}_{L^{4}(T)}.\) 

Hence \(H_1, H_2\)  and \(H_3\)  are bounded linear functionals of the variable \(\psi \)  as in (REF ) with \(\psi _x, \psi _t \in L^2(D)\)  and \(\psi _t(0,\cdot )\in {L^{4}(T)}\) . For such \(\psi \)  we use the above formulae for \(H_1, H_2, H_3\)  and compute the linear combination
\(-H_1(\psi )+H_2(\psi )-\gamma H_3(\psi ) = -2\sum _{k} \frac{\hat{\alpha }_k}{k^2} \Phi _k^{\prime }(0)\Psi _k(0) - \frac{\gamma \omega ^4}{T} (\hat{\alpha }*\hat{\alpha }*\hat{\alpha }*\hat{y})_0=0\) 

due to the Euler-Lagrange equation for the functional \(J\) , i.e., the vanishing of \(J^{\prime }(\hat{\alpha })[\hat{y}]\)  in (REF ) for all \(\hat{y} \in {\mathcal {D}(J)}\) . The last equality means that \(w\)  is a weak solution of (REF ).
